[GIF] Fall In (Stereographic projection of dots on the sphere)

This is fairly straightforward: put 32 equally-spaced dots of (spherical) radius 0.03 on 20 equally-spaced circles of latitude on the unit sphere. Move the circles steadily down (with constant speed in the
$z$-direction, not constant speed on the sphere), and project the whole image to the plane, adding in a color gradient. The key function is ProjectedSphericalCircle[], which converts a spherical disk to a stereographically-projected planar disk; the code for that function is here.

One slight nuisance is that Disk[] doesn't seem to rasterize very well at low resolutions: the positioning of the disk in the resulting image is not very precise. I presume this is due to the discreteness of the pixel grid, and that the rasterization function tries to center the disk on a pixel or something of that nature, but the result is that outputting this to a GIF at the final resolution produces a bunch of dots that dance around quite a bit, as in this version:

To get smooth movement I ended up having to output at 2160x2160 and then resize in ImageMagick to the final 540x540 resolution.